curve #1842

y² = x³ + x² − 69033910830418349762773631457605701396585736530025x + 224639114783370852423625209868649047871661605502504267080975248576709215419

a-invariants: [0, 1, 0, -69033910830418349762773631457605701396585736530025, 224639114783370852423625209868649047871661605502504267080975248576709215419]
rank (lower bound): ≥ 29
torsion subgroup: trivial
conductor (N): 3460236869446370518380164296016295807463617217764288910197137420940152152328816761301079023715063694153350125601768228129208469271773688
discriminant (Δ): -744310642788826190109855484743568891337054337372606035243742150467537321036815689355352789143288088156765867523574938161840008764187993232337117235968
Faltings height: 27.5494
naive height: 355.9244
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 79, 53777, 104962538490183553082931059871240076408687434519327647122631653982954536921624171054693532477239384056504158090864647083793
regulator: 178937238015533733691291569353162407.765854851217694903590274443956135311807195840145470394049327
submitted by: Ritik Jain
submitted at: 2026-09-25 17:05:36 UTC
last updated: 2026-09-25 17:05:36 UTC

Witness: 29 independent points log in to add more points →

(957678902540559740340319963581266228548685225874901381/15580406729983896377710292041, 929050613775621193731784307407080591184237627802538243558713779868155300992029646/1944769730391858247623567255260836860545339)
(2673799071760626931861130488098782225131/977293699319809, -229132519073239781425572118199185791736118196161955292614954/30551859508144975055873)
(1293787876342305400538299573/9, -46460564180029599794145040239143945547902/27)
(106763167025233457160913099/9, -885265953772879201934823488201304157022/27)
(66028302010983494133033277, -532477658219040223668673990358129082966)
(-7226743085188191864059213, 18603948522434538515020317850506346746)
(6091497972875442294972009913/16, -475317270802893141220108619446439052364875/64)
(1288962593425353944500025209/64, -42847766753667948063637074863944436261907/512)
(8069304753702791571089543941/841, 514827949678479046082657011376768831429454/24389)
(2985635248543714284430339, -6718857278182044432468678821241560922)
(1861971208419415982172056899/25, 79865571320106991331844148175887768136418/125)
(6830321441211770228223781, 8471852643999743672298395481326201586)
(23793231609357399856688806/9, -210204572057261440931206510470964987379/27)
(-6366243222324953486899250, -20152120837286391011346350993113497237)
(2981269017716850820871767, 6732597552704899534270002877536178986)
(1658485359666429380907692590/9, -67473521653751444499212759661169717502449/27)
(3005597682019581299402727871717449319/179792112361, 4663677072502829157753840396909562961005540088520129762/76235271691198859)
(8383316607025073009070975999322717/1907855041, 206473568610443072553318147734230578771460391494794/83333200335839)
(-20353167184580490594667150361141096357008801475942586013538442060891940386770458543039135902629568908974366941039958275005273/3195612614231349235691939268940184128356289669353813393236621704708094869131561928863019405560745489, 3639750835633385736611173559781633200668858529512699677284584055852793901975738883132372500053156991474577930855069822336688448459739277440802401856799194070148955857218919119159430291430/180647181590554744875630672307163117182662700260417467852211231590674783398984741993881468607155698164684398607271987303548038176957435099158029103737)
(64824123288526308010808142898314145335290083873498863105165169787423874694573830790/15287016499641711202453176681317181956337249007476754092169, 5396971302063420415733569791164580074443381112459253665302408179850206705240281556107186479614307244740480725840768791752989/1890097037836037281231715327920355806690499621705740882124099788962810911736898897203803)
(3614824879291958221277453737552927090737349339732922277584748942247769256343199091517284670398694801339726130717815/1028488153073251687585191004953256625018552197158288102718198509545573468883257792157234241, 5259131824511883479350737249202408048032334886183332691552660756918646703502916261401801303418677134093732627656968502883204549745396562468857304037360479367105711402258218/1043035140388534369864435011353412027153747175615506225667720213669707740775195963968408420461781275312406497199056209882479267981988961)
(1913781675841255972365297251392934484222217392928726682127682690261/383278507900254171828726031126580105056356, 499407121439074548130789606361653545391483930715447238208317589509515367797495069704159767138997969/237285773450128077158353389775060212138761013058986757998185496)
(82024377105796830007438332455751093898698248481263189701941047414246460036885456174622/14346969423614056904401687799300948437356734807745607272197329, 222955311268364148373404810251447068880357550877476228104235949383386254989428493968172089299886546809305645013824868908139211115/54342580714967811413829948115131063515177876434983373283194127967712504770780933774048881767)
(10994261009508669132630903371042371974927987236663477869032118/2171244586819387024764720420944209849, 7089122695501612448760514149403746660813449476052216368193921608351837819249964885451469533/3199360090786209091766660839465730219135079750229651757)
(-25823516264897041541358912346959186634575964484163995040772242249/2713473243077800424079650182919214909121, 627276050625926116258837579338956688315813636204153910260898015758414579597602078129473834506966/141347559762969716129466288879550863732513798181494564093919)
(-8631555448444518057979290761364946511782127024784101929162927950717180601931/1286903651333555129389911967620923823455336468529369, 906926581461304779976405620925759785241815946520290295251138917616242621973756834595051386286310037676351541070658/46165660095011635305039955322284860181007006039758970917431204635373032301597)
(5178136747992439025087060928153587851761560916318504007729842329503622304815091964440993418681856049882498218876758868987419274999884908789731167150/1366316510710636431294707976166338191805247175248887499246833136843724904929956598611585937069142001285560798672858746885161, 210937979143536721398867957296567275746796184170871044429763569198633008921501884979614732166774157646590764754442257179787109598483995394383911000260664219908266787685356240162676159222086086924402647628897805885892293567/50504139352572887889529861655863384607423442589221570307504776737097330361174889392867668065124842215495641978895973360994511379779215251373661319064861156396564446947352864083832783659)
(-1068537318885120847767934197561698957029733108610308395719675584759443304096728105170254201529/113731265601098230698924652528094089779819417401167802723724617294401, 8036058373703439448662583801738833925846096441539003382440216446506760232467595929888328270007825060828058327344911196189481255691126363786/1212885510864358437798781707462780298918462717911632745595193330045221395623384836136464975555396298399)
(-1342255604448885370453930946882554687009767290370947924026261989076135363219331458989490781015320445315/257173533699686677216535034689291377214350753825973008038539614440614080671649, 2744277032389465277515977400649342824347491541162176036824055044974332131156959607338057149506244127968928328343312812775675712453455297281220636787839466/130418562348574978382006031400905216096720218886785411629495032280491562491923649618196802203343881963049373453167857)

Commentary

Specialization at T = 5523/1265 of the rank-17 fibration of X1092 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 147997T^2 - 120070T - 74711, p = 3738T^2 - 33976T - 778384, q = 69(2277T^2 + 2469T - 794), D = 592091556T^4 - 877088375T^3 - 81950180265T^2 - 578365613920T - 97926171696, E = 12455943192T^4 - 9463234483T^3 - 280485479886T^2 - 355911161357T + 8935370934.

last edited by wgxli at 2026-09-25 17:55:34 UTC · history